Split a Total with Reciprocal Condition Divide 50 into two positive parts such that the sum of their reciprocals equals 1/12. What are the two parts?

Introduction / Context:
This is a classic two-part split constrained by a condition on reciprocals. The key is to convert the verbal condition into an algebraic equation in a single variable and solve a simple quadratic. Such questions appear frequently in number system sections of aptitude tests.

Given Data / Assumptions:

• Total sum of the two parts is 50.
• Let the parts be x and 50 − x, with x > 0 and 50 − x > 0.
• 1/x + 1/(50 − x) = 1/12.

Concept / Approach:
Add the reciprocals using a common denominator and equate to 1/12. This yields a quadratic equation in x. Solve the quadratic to obtain two positive solutions, which are complementary parts summing to 50. Check both solutions for validity and verify the reciprocal condition numerically.

Step-by-Step Solution:
Start: 1/x + 1/(50 − x) = (50) / (x(50 − x)) = 1/12.Therefore, x(50 − x) = 600.Expand: 50x − x^2 = 600 → x^2 − 50x + 600 = 0.Solve: Discriminant D = 2500 − 2400 = 100; x = (50 ± 10)/2 → x = 20 or 30.

Verification / Alternative check:
Check 20 and 30: 1/20 + 1/30 = (3 + 2)/60 = 5/60 = 1/12. Condition satisfied exactly.

Why Other Options Are Wrong:
24, 26 give 1/24 + 1/26 ≠ 1/12; similarly 28, 22 and 36, 14 and 18, 32 do not satisfy the reciprocal sum constraint.

Common Pitfalls:
Algebra sign errors when forming the quadratic; forgetting both roots are valid and symmetric around 25; mixing reciprocal sum with arithmetic sum or difference.

Final Answer:
20, 30